Injection molding is probably the molding technique of choice because it has the highest density (lowest voids), higher tensile strength plastics, and greatest detail (like a plastic model kit). However, there are many, many other ways to to do this and they depend on your application and the number of parts you want to make as well as what material you need to use.
A company needs a good mold maker, injection molding machines, skill, money, and some luck. Molds are typically machined out of steel with injection ports for the molten plastic and relief ports for outgoing air. Usually, molds are a two-piece steel mold that clamps together to form a 3-D cavity, but can be made out of other materials, including ceramic. Hot, molten plastic is injected into the mold, the plastic cools and solidifies, the mold is opened, and the part ejected and finished.
There is a lot more details to the process than what I described, but essentially, it is a matter of heating plastic until it melts, pumping it into the mold and releasing it from the mold. Finally, any burs and flash is removed and any finishing operations completed to make the part a finished product.
There are many, many companies that specialize in doing this. If you have none of the above (and it sounds like you don't) it probably would be best to start with a mold making machine shop or one of the many injection molding companies that specialize in the process. They will guide you through their process and help you design your part.
There will be tooling fees to make the molds and there are different types of molds and quality of molds, too. A molding company will help you select the process that best fits your needs and budget and produce the parts for you to your specifications.